Hierarchical clustering has usually been addressed by discrete optimization using heuristics or continuous optimization of relaxed scores for hierarchies. In this work, we propose to optimize expected scores under a probabilistic model over hierarchies. (1) We show theoretically that the global optimal values of the expected Dasgupta cost and Tree-Sampling divergence (TSD), two unsupervised metrics for hierarchical clustering, are equal to the optimal values of their discrete counterparts contrary to some relaxed scores. (2) We propose Expected Probabilistic Hierarchies (EPH), a probabilistic model to learn hierarchies in data by optimizing expected scores. EPH uses differentiable hierarchy sampling enabling end-to-end gradient descent based optimization, and an unbiased subgraph sampling approach to scale to large datasets. (3) We evaluate EPH on synthetic and real-world datasets including vector and graph datasets. EPH outperforms all other approaches quantitatively and provides meaningful hierarchies in qualitative evaluations.

Despite its success in the image domain, adversarial training did not (yet) stand out as an effective defense for Graph Neural Networks (GNNs) against graph structure perturbations. In the pursuit of fixing adversarial training (1) we show and overcome fundamental theoretical as well as practical limitations of the adopted graph learning setting in prior work; (2) we reveal that flexible GNNs based on learnable graph diffusion are able to adjust to adversarial perturbations, while the learned message passing scheme is naturally interpretable; (3) we introduce the first attack for structure perturbations that, while targeting multiple nodes at once, is capable of handling global (graph-level) as well as local (node-level) constraints. Including these contributions, we demonstrate that adversarial training is a state-of-the-art defense against adversarial structure perturbations.

Pruning, the task of sparsifying deep neural networks, received increasing attention recently. Although state-of-the-art pruning methods extract highly sparse models, they neglect two main challenges: (1) the process of finding these sparse models is often very expensive; (2) unstructured pruning does not provide benefits in terms of GPU memory, training time, or carbon emissions. We propose Early Compression via Gradient Flow Preservation (EarlyCroP), which efficiently extracts state-of-the-art sparse models before or early in training addressing challenge (1), and can be applied in a structured manner addressing challenge (2). This enables us to train sparse networks on commodity GPUs whose dense versions would be too large, thereby saving costs and reducing hardware requirements. We empirically show that EarlyCroP outperforms a rich set of baselines for many tasks (incl. classification, regression) and domains (incl. computer vision, natural language processing, and reinforcment learning). EarlyCroP leads to accuracy comparable to dense training while outperforming pruning baselines.

Dirichlet-based uncertainty (DBU) models are a recent and promising class of uncertainty-aware models. DBU models predict the parameters of a Dirichlet distribution to provide fast, high-quality uncertainty estimates alongside with class predictions. In this work, we present the first large-scale, in-depth study of the robustness of DBU models under adversarial attacks. Our results suggest that uncertainty estimates of DBU models are not robust w.r.t. three important tasks: (1) indicating correctly and wrongly classified samples; (2) detecting adversarial examples; and (3) distinguishing between in-distribution (ID) and out-of-distribution (OOD) data. Additionally, we explore the first approaches to make DBU mod- els more robust. While adversarial training has a minor effect, our median smoothing based ap- proach significantly increases robustness of DBU models.

Graph Neural Networks (GNNs) are increasingly important given their popularity and the diversity of applications. Yet, existing studies of their vulnerability to adversarial attacks rely on relatively small graphs. We address this gap and study how to attack and defend GNNs at scale. We propose two sparsity-aware first-order optimization attacks that maintain an efficient representation despite optimizing over a number of parameters which is quadratic in the number of nodes. We show that common surrogate losses are not well-suited for global attacks on GNNs. Our alternatives can double the attack strength. Moreover, to improve GNNs' reliability we design a robust aggregation function, Soft Median, resulting in an effective defense at all scales. We evaluate our attacks and defense with standard GNNs on graphs more than 100 times larger compared to previous work. We even scale one order of magnitude further by extending our techniques to a scalable GNN.

The interdependence between nodes in graphs is key to improve class prediction on nodes, utilized in approaches like Label Probagation (LP) or in Graph Neural Networks (GNNs). Nonetheless, uncertainty estimation for non-independent node-level predictions is under-explored. In this work, we explore uncertainty quantification for node classification in three ways: (1) We derive three axioms explicitly characterizing the expected predictive uncertainty behavior in homophilic attributed graphs.(2) We propose a new model Graph Posterior Network (GPN) which explicitly performs Bayesian posterior updates for predictions on interdependent nodes. GPN provably obeys the proposed axioms. (3) We extensively evaluate GPN and a strong set of baselines on semi-supervised node classification including detection of anomalous features, and detection of left-out classes. GPN outperforms existing approaches for uncertainty estimation in the experiments.

The study of vertex centrality measures is a key aspect of network analysis. Naturally, such centrality measures have been generalized to groups of vertices; for popular measures it was shown that the problem of finding the most central group is -hard. As a result, approximation algorithms to maximize group centralities were introduced recently. Despite a nearly-linear running time, approximation algorithms for group betweenness and (to a lesser extent) group closeness are rather slow on large networks due to high constant overheads. That is why we introduce GED-Walk centrality, a new submodular group centrality measure inspired by Katz centrality. In contrast to closeness and betweenness, it considers walks of any length rather than shortest paths, with shorter walks having a higher contribution. We define algorithms that (i) efficiently approximate the GED-Walk score of a given group and (ii) efficiently approximate the (proved to be -hard) problem of finding a group with highest GED-Walk score. Experiments on several real-world datasets show that scores obtained by GED-Walk improve performance on common graph mining tasks such as collective classification and graph-level classification. An evaluation of empirical running times demonstrates that maximizing GED-Walk (in approximation) is two orders of magnitude faster compared to group betweenness approximation and for group sizes ≤ 100 one to two orders faster than group closeness approximation. For graphs with tens of millions of edges, approximate GED-Walk maximization typically needs less than one minute. Furthermore, our experiments suggest that the maximization algorithms scale linearly with the size of the input graph and the size of the group.

Recent works show that message-passing neural networks (MPNNs) can be fooled by adversarial attacks on both the node attributes and the graph structure. Since MPNNs are currently being rapidly adopted in real-world applications, it is thus crucial to improve their reliablility and robustness. While there has been progress on robustness certification of MPNNs under perturbation of the node attributes, no existing method can handle structural perturbations. These perturbations are especially challenging because they alter the message passing scheme itself. In this work we close this gap and propose the first method to certify robustness of Graph Convolutional Networks (GCNs) under perturbations of the graph structure. We show how this problem can be expressed as a jointly constrained bilinear program - a challenging, yet well-studied class of problems - and propose a novel branch-and-bound algorithm to obtain lower bounds on the global optimum. These lower bounds are significantly tighter and can certify up to twice as many nodes compared to a standard linear relaxation.

Accurate estimation of aleatoric and epistemic uncertainty is crucial to build safe and reliable systems. Traditional approaches, such as dropout and ensemble methods, estimate uncertainty by sampling probability predictions from different submodels, which leads to slow uncertainty estimation at inference time. Recent works address this drawback by directly predicting parameters of prior distributions over the probability predictions with a neural network. While this approach has demonstrated accurate uncertainty estimation, it requires defining arbitrary target parameters for in-distribution data and makes the unrealistic assumption that out-of-distribution (OOD) data is known at training time. In this work we propose the Posterior Network (PostNet), which uses Normalizing Flows to predict an individual closed-form posterior distribution over predicted probabilites for any input sample. The posterior distributions learned by PostNet accurately reflect uncertainty for in- and out-of-distribution data -- without requiring access to OOD data at training time. PostNet achieves state-of-the art results in OOD detection and in uncertainty calibration under dataset shifts.

Perturbations targeting the graph structure have proven to be extremely effective in reducing the performance of Graph Neural Networks (GNNs), and traditional defenses such as adversarial training do not seem to be able to improve robustness. This work is motivated by the observation that adversarially injected edges effectively can be viewed as additional samples to a node's neighborhood aggregation function, which results in distorted aggregations accumulating over the layers. Conventional GNN aggregation functions, such as a sum or mean, can be distorted arbitrarily by a single outlier. We propose a robust aggregation function motivated by the field of robust statistics. Our approach exhibits the largest possible breakdown point of 0.5, which means that the bias of the aggregation is bounded as long as the fraction of adversarial edges of a node is less than 50%. Our novel aggregation function, Soft Medoid, is a fully differentiable generalization of the Medoid and therefore lends itself well for end-to-end deep learning. Equipping a GNN with our aggregation improves the robustness with respect to structure perturbations on Cora ML by a factor of 3 (and 5.5 on Citeseer) and by a factor of 8 for low-degree nodes.

Deep learning models for graphs have achieved strong performance for the task of node classification. Despite their proliferation, little is known about their robustness to adversarial attacks. Yet, in domains where they are likely to be used, e.g., the web, adversaries are common. Can deep learning models for graphs be easily fooled? In this work, we present a study of adversarial attacks on attributed graphs, specifically focusing on models exploiting ideas of graph convolutions. In addition to attacks at test time, we tackle the more challenging class of poisoning/causative attacks, which focus on the training phase of a machine learning model. We generate adversarial perturbations targeting the node’s features and the graph structure , thus, taking the dependencies between instances in account. Moreover, we ensure that the perturbations remain unnoticeable by preserving important data characteristics. To cope with the underlying discrete domain, we propose an efficient algorithm N ettack exploiting incremental computations. Our experimental study shows that accuracy of node classification significantly drops even when performing only few perturbations. Even more, our attacks are transferable: the learned attacks generalize to other state-of-the-art node classification models and unsupervised approaches, and likewise are successful even when only limited knowledge about the graph is given. For the first time, we successfully identify important patterns of adversarial attacks on graph neural networks (GNNs) — a first step towards being able to detect adversarial attacks on GNNs.

Deep learning models for graphs have achieved strong performance for the task of node classification. Despite their proliferation, currently there is no study of their robustness to adversarial attacks. Yet, in domains where they are likely to be used, e.g. the web, adversaries are common. Can deep learning models for graphs be easily fooled? In this extended abstract we summarize the key findings and contributions of our work, in which we introduce the first study of adversarial attacks on attributed graphs, specifically focusing on models exploiting ideas of graph convolutions. In addition to attacks at test time, we tackle the more challenging class of poisoning/causative attacks, which focus on the training phase of a machine learning model. We generate adversarial perturbations targeting the node's features and the graph structure, thus, taking the dependencies between instances in account. Moreover, we ensure that the perturbations remain unnoticeable by preserving important data characteristics. To cope with the underlying discrete domain we propose an efficient algorithm Nettack exploiting incremental computations. Our experimental study shows that accuracy of node classification significantly drops even when performing only few perturbations. Even more, our attacks are transferable: the learned attacks generalize to other state-of-the-art node classification models and unsupervised approaches, and likewise are successful given only limited knowledge about the graph.

Recent works show that Graph Neural Networks (GNNs) are highly non-robust with respect to adversarial attacks on both the graph structure and the node attributes, making their outcomes unreliable. We propose the first method for certifiable (non-)robustness of graph convolutional networks with respect to perturbations of the node attributes. We consider the case of binary node attributes (e.g. bag-of-words) and perturbations that are L_0-bounded. If a node has been certified with our method, it is guaranteed to be robust under any possible perturbation given the attack model. Likewise, we can certify non-robustness. Finally, we propose a robust semi-supervised training procedure that treats the labeled and unlabeled nodes jointly. As shown in our experimental evaluation, our method significantly improves the robustness of the GNN with only minimal effect on the predictive accuracy.

We propose NetGAN - the first implicit generative model for graphs able to mimic real-world networks. We pose the problem of graph generation as learning the distribution of biased random walks over the input graph. The proposed model is based on a stochastic neural network that generates discrete output samples and is trained using the Wasserstein GAN objective. NetGAN is able to produce graphs that exhibit well-known network patterns without explicitly specifying them in the model definition. At the same time, our model exhibits strong generalization properties, as highlighted by its competitive link prediction performance, despite not being trained specifically for this task. Being the first approach to combine both of these desirable properties, NetGAN opens exciting avenues for further research.

Deep learning models for graphs have achieved strong performance for the task of node classification. Despite their proliferation, currently there is no study of their robustness to adversarial attacks. Yet, in domains where they are likely to be used, e.g. the web, adversaries are common. Can deep learning models for graphs be easily fooled? In this work, we introduce the first study of adversarial attacks on attributed graphs, specifically focusing on models exploiting ideas of graph convolutions. In addition to attacks at test time, we tackle the more challenging class of poisoning/causative attacks, which focus on the training phase of a machine learning model.We generate adversarial perturbations targeting the node's features and the graph structure, thus, taking the dependencies between instances in account. Moreover, we ensure that the perturbations remain unnoticeable by preserving important data characteristics. To cope with the underlying discrete domain we propose an efficient algorithm Nettack exploiting incremental computations. Our experimental study shows that accuracy of node classification significantly drops even when performing only few perturbations. Even more, our attacks are transferable: the learned attacks generalize to other state-of-the-art node classification models and unsupervised approaches, and likewise are successful even when only limited knowledge about the graph is given.